First of all, is this an azalea? Sorry for posting here if it isn't.
Second, if it is an azalea, what is this disease and how can it be treated?
They gray-green layered feathery growths on the bark are lichens. They aren't harmful, and can be found on healthy plants, but more often on plants that have been weakened or are under stress by something else.
You've got a lot of dead wood in that azalea, it should be removed so as not to offer a foothold for secondary insect or disease problems. Is it watered regularly in our dry late summer and fall? A mulch of the root zone to cool roots and conserve moisture would be helpful...if you were to top dress with a couple of inches of compost, you would be conditioning the soil and adding nutrients too. (you can buy a very well composted steer manure for $1 bag at Home Depot)
You were trying to show the lichen and not the foliage, but I wonder if I'm not seeing leaf gall in your photos, pretty common with our wet mild Springs. It's too late to do anything but remove the galls this year, if you do have them, a fungicide spray next Spring before new growth buds open would be helpful....
Here is a link that might be useful: Leaf gall
Thank you morz8. We just moved in a month ago, so I don't know how that plant was watered (IF it was watered). In that particular bed, the former owners put down some sort of fabric covering to keep down the weeds (which manage to grow in the thin layer of soil on top of the fabric). It drives me nuts when I want to plant something else. It's a big area, but I'm thinking of carefully cutting/pulling it all up and using mulch.
The dead wood isn't necessarily the branches with the lichens is it?
No, lichens will grown on live wood, in fact, on live healthy wood. Or dead wood. I have them on a cedar fence and find them kind of interesting there :)
I'm really not a fan of landscape fabric for the reasons you mention and I especially don't like it over shallow rooted rhododendrons, azaleas.